Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 22 / 2 / c / Solution

The representing object in the functor category $[J,\mathcal{C}]$ is the constant diagram in a category ${\Delta }_{J}C$. If $p_j:{\lim }_j{E}_j\to E_j$ is the chosen categorical limit cone, the natural bijection isA natural transformation from the constant diagram in a category is precisely a categorical cone with vertex $C$, and its inverse is the unique mediating morphism from the universal property of the categorical limit. For $\alpha :E\Rightarrow E^{\prime }$, the defining equations for $\lim \alpha$ show that postcomposition by $\lim \alpha$ corresponds to postcomposition by $\alpha$ on the right. This establishes naturality in $E$ and proves representability by the constant diagram.